Finite Group Databasebeta

Subgroups — #10617381

Order 640.

322 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66062	C₂	1	1	yes	maximal, characteristic
2	#65788	C₂	1	2	yes	maximal, complemented
3	Q₈×#170	C₂	1	2	yes	maximal, complemented
4	#65707	C₂	1	4	yes	maximal, complemented
5	#65706	C₂	1	4	yes	maximal
6	#65683	C₂	1	1	yes	maximal, characteristic
7	#65678	C₂	1	1	yes	maximal, characteristic
8	C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	#4092	C₄	1	2	yes	—
10	Q₈×Dic₅	C₂²	1	2	yes	—
11	#4043	C₄	1	2	yes	—
12	#4038	C₂²	1	2	yes	—
13	#4036	C₄	1	2	yes	—
14	#4033	C₂²	1	2	yes	—
15	C₄²×D₅	C₂²	1	1	yes	characteristic
16	#4015	C₄	1	1	yes	characteristic
17	#4004	C₂²	1	2	yes	—
18	#4004	C₂²	1	4	yes	—
19	#4003	C₂²	1	2	yes	—
20	#4003	C₂²	1	2	yes	—
21	#4002	C₂²	1	2	yes	—
22	#4002	C₂²	1	4	yes	—
23	C₄⋊#170	C₂²	1	2	yes	—
24	C₄⋊#170	C₂²	1	4	yes	—
25	C₄×#170	C₂²	1	1	yes	characteristic
26	C₄×#170	C₂²	1	1	yes	characteristic
27	C₄×#170	C₂²	1	4	yes	—
28	#2985	—	5	1	no	maximal, Sylow
29	C₂×C₅×Q₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
34	C₂×#170	C₂³	1	2	yes	—
35	C₂×#170	C₂³	1	4	yes	—
36	C₂×#170	C₂³	1	2	yes	—
37	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	#685	C₂×C₄	1	4	yes	—
41	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₅	C₂×C₄	1	4	yes	—
44	#683	C₂×C₄	1	4	yes	—
45	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
46	C₄×Dic₅	C₂³	1	1	yes	characteristic
47	C₄×Dic₅	C₂³	1	4	yes	—
48	C₄×Dic₅	C₂³	1	1	yes	characteristic
49	#533	—	5	1	no	—
50	#446	—	5	4	no	—
51	#446	—	5	2	no	—
52	C₈×Q₈	—	5	2	no	—
53	#433	—	5	4	no	—
54	#432	—	5	1	no	—
55	#431	—	5	1	no	—
56	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
57	C₅×Q₈	—	2	4	no	—
58	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
64	C₂×Dic₅	C₂²×C₄	1	4	yes	—
65	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
66	C₄×D₅	—	2	2	no	—
67	C₄×D₅	—	2	2	no	—
68	C₄×D₅	—	4	1	no	—
69	Dic₁₀	—	4	2	no	—
70	#170	—	2	2	no	—
71	#170	—	2	4	no	—
72	#170	—	2	2	no	—
73	C₄⋊Q₈	—	5	1	no	—
74	C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
75	#125	—	5	2	no	—
76	C₂²⋊Q₈	—	5	2	no	—
77	C₂²⋊Q₈	—	5	2	no	—
78	C₄×Q₈	—	5	2	no	—
79	C₄×Q₈	—	5	2	no	—
80	#117	—	5	2	no	—
81	C₂×C₄²	—	5	1	no	—
82	C₄⋊C₈	—	5	2	no	—
83	C₄⋊C₈	—	5	4	no	—
84	C₄⋊C₈	—	5	2	no	—
85	C₄⋊C₈	—	5	4	no	—
86	C₂²⋊C₈	—	5	2	no	—
87	C₂²⋊C₈	—	5	2	no	—
88	#97	—	5	2	no	—
89	#97	—	5	4	no	—
90	C₄×C₈	—	5	1	no	—
91	C₄×C₈	—	5	1	no	—
92	C₄×C₈	—	5	4	no	—
93	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₅	—	2	1	no	—
95	C₂×D₅	—	2	1	no	—
96	C₂×D₅	—	2	1	no	—
97	C₄×C₅	—	2	1	no	—
98	C₄×C₅	—	2	4	no	—
99	C₄×C₅	—	2	1	no	—
100	C₄×C₅	#131	1	2	yes	—
101	Dic₅	—	2	1	no	—
102	Dic₅	—	2	4	no	—
103	Dic₅	—	2	1	no	—
104	Dic₅	2₋¹⁺⁴	1	2	yes	—
105	C₂×Q₈	—	5	2	no	—
106	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
107	C₂²×C₄	—	5	1	no	—
108	C₂²×C₄	—	5	1	no	—
109	C₂²×C₄	—	5	1	no	—
110	C₂×C₈	—	5	2	no	—
111	C₂×C₈	—	5	2	no	—
112	C₂×C₈	—	5	4	no	—
113	C₄⋊C₄	—	5	2	no	—
114	C₄⋊C₄	—	5	4	no	—
115	C₄⋊C₄	—	5	4	no	—
116	C₄⋊C₄	—	5	2	no	—
117	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
118	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
119	C₂²⋊C₄	—	5	4	no	—
120	C₄²	—	5	1	no	—
121	C₄²	—	5	4	no	—
122	C₄²	—	5	1	no	—
123	C₄²	—	5	1	no	—
124	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₂×C₅	#519	1	1	yes	characteristic
126	C₂×C₅	C₂×#131	1	1	yes	characteristic
127	C₂×C₅	#568	1	1	yes	characteristic
128	D₅	—	4	1	no	—
129	C₂³	—	5	1	no	—
130	Q₈	—	2	4	no	—
131	Q₈	—	20	2	no	—
132	C₂×C₄	—	5	1	no	—
133	C₂×C₄	—	5	1	no	—
134	C₂×C₄	—	5	1	no	—
135	C₂×C₄	—	5	4	no	—
136	C₂×C₄	—	10	2	no	—
137	C₂×C₄	—	10	2	no	—
138	C₂×C₄	—	20	1	no	—
139	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
142	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
143	C₈	—	10	2	no	—
144	C₈	—	10	2	no	—
145	C₈	—	10	4	no	—
146	C₅	#2985	1	1	yes	characteristic, complemented, Sylow
147	C₂²	—	10	1	no	—
148	C₂²	—	10	1	no	—
149	C₂²	—	10	1	no	—
150	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
151	C₄	—	2	4	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	5	2	no	—
155	C₄	—	10	1	no	—
156	C₄	—	10	1	no	—
157	C₄	—	10	4	no	—
158	C₄	#4131	1	2	yes	—
159	C₂	—	20	1	no	—
160	C₂	C₂×#4131	1	1	yes	characteristic
161	C₂	#66260	1	1	yes	characteristic
162	C₂	#66262	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×C₄⋊Q₈	C₄
prime radical	C₂²×D₅	C₂²×C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	C₄⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#2985
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #65788	C₂	semidirect
3. Q₈×#170	C₂	semidirect
4. #65707	C₂	semidirect
146. C₅	#2985	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#66062	2
#65788	2
Q₈×#170	2
#65707	2
#65706	2
#65683	2
#65678	2
#2985	5